An iterative algorithm to compute the Bott-Duffin inverse and generalized Bott-Duffin inverse

Let L be a subspace of Cn and PL be the orthogonal projector of Cn onto L. For A?Cn?n, the generalized Bott-Duffin (B-D) inverse A(+)(L) is given by A(+)(L)= PL(APL + PL?)?. In this paper, by defined a non-standard inner product, a finite formulae is presented to compute Bott-Duffin inverse A(?)(L) = PL(APL+P?)? and generalized Bott-Duffin inverse A(?)(L)= PL (APL+PL?)? under the condition A is L?zero (i.e., AL?L?={0}). By this iterative method, when taken the initial matrix X0 = PLA?PL, the Bott-Duffin inverse A(?1)(L) and generalized Bott-duffin inverse A(?)(L) can be obtained within a finite number of iterations in absence of roundoff errors. Finally a given numerical example illustrates that the iterative algorithm dose converge.